Circulation and Vorticity

Consider a closed curve $C$ situated entirely within a moving fluid. The vector line integral (see Section A.14)

$${\mit\Gamma}_C = \oint_C {\bf v}\cdot d{\bf r},$$ (4.76)

where $d{\bf r}$ is an element of $C$ , and the integral is taken around the whole curve, is termed the circulation of the flow around the curve. The sense of circulation (i.e., either clockwise or counter-clockwise) is arbitrary.

Let $S$ be a surface having the closed curve $C$ for a boundary, and let $d{\bf S}$ be an element of this surface (see Section A.7) with that direction of the normal which is related to the chosen sense of circulation around $C$ by the right-hand circulation rule. (See Section A.8.) According to the curl theorem (see Section A.22),

$${\mit\Gamma}_C= \oint_C {\bf v}\cdot d{\bf r} = \int_S \mbox{\boldmath$\omega$} \cdot d{\bf S}.$$ (4.77)

Thus, we conclude that circulation and vorticity are intimately related to one another. In fact, according to the previous expression, the circulation of the fluid around loop $C$ is equal to the net sum of the intensities of the vortex filaments passing through the loop and piercing the surface $S$ (with a filament making a positive, or negative, contribution to the sum depending on whether it pierces the surface in the direction determined by the chosen sense of circulation around $C$ and the right-hand circulation rule, or in the opposite direction). One important proviso to Equation (4.77) is that the surface $S$ must lie entirely within the fluid.